Problem: At Gauss Middle School, the current student-to-teacher ratio is $17.5:1$. The school currently has $560$ students. Next year, the student-to-teacher ratio must be $19:1$. If the school gains $10$ students, how many fewer teachers will be needed next year?
Answer: Right now, there is one teacher for every 17.5 students, so there must be $\frac{560}{17.5}=32$ teachers at Gauss Middle School. If the school gains 10 students and changes the ratio, they will need $\frac{570}{19}=30$ teachers next year. Thus, they will need $32-30=\boxed{2}$ fewer teachers.